Constraints Satisfaction Edmondo Trentin DIISM Constraint Satisfaction Problems

Constraints Satisfaction Edmondo Trentin, DIISM

Constraint Satisfaction Problems: Local Search • In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution • State space = set of "complete" configurations • Problem: find a configuration (i. e. , a state) satisfying some constraints (e. g. , n-queens) • In such cases, we can use local search algorithms • The general idea: keep a single "current" state, try to improve it by application of a single “expansion”

Example: n-queens • Put n queens on an n × n board with no two queens on the same row, column, or diagonal

Hill-climbing search • Basically, a discrete-step instance of the gradient method • Requires an objective (criterion) function

Hill-climbing search • Problem: depending on initial state, can get stuck in local maxima

Hill-climbing search: 8 -queens problem • • • h = number of pairs of queens that are attacking each other, either directly or indirectly h is our criterion function and, in this case, it has to be minimized h = 17 for the above state

Hill-climbing search: 8 -queens problem • A local minimum with h = 1

Simulated annealing search • Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency

Properties of simulated annealing search • One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1

Local beam search • Keep track of k states rather than just one • Start with k randomly generated states • At each iteration, all the successors of all k states are generated • If any one is a goal state, stop; else select the k best successors from the complete list and repeat.

Genetic algorithms • A successor state is generated by combining two parent states • Start with k randomly generated states (population) • A state is represented as a string over a finite alphabet (often a string of 0 s and 1 s) • Evaluation function (fitness function). Higher values for better states. • Produce the next generation of states by selection, crossover, and mutation

Genetic algorithms • Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28) • 24/(24+23+20+11) = 31% • 23/(24+23+20+11) = 29% etc

Genetic algorithms